Entropy-Stable, High-Order Discretizations Using Continuous Summation-By-Parts Operators

We present summation-by-parts (SBP) discretizations of the linear advection and Euler equations that use a continuous solution space. These continuous SBP discretizations assemble global operators from element operators, and they are the SBP analog, or generalization, of continuous Galerkin finite-element methods. Consequently, a stabilization method is needed to suppress high-frequency oscillations and ensure optimal convergence rates. The proposed stabilization, which is a form of local-projection stabilization, leads to energy-stable discretizations of the linear advection equation and entropy-stable discretizations of the Euler equations. Furthermore, the stabilization is element local, which keeps the stencil compact, and the stabilized discretizations have favorable time-step restrictions. Results are provided to demonstrate the accuracy and efficiency of the continuous SBP discretizations.